Dehn's lemma

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In mathematics Dehn's lemma asserts that a piecewise-linear map of a disk into a 3-manifold, with the map's singularity set in the disc's interior, implies the existence of another piecewise-linear map of the disc which is an embedding and is identical to the original on the boundary of the disc.

This theorem was thought to be proven by Max Dehn in 1910, but an error was later found by Hellmuth Kneser. The status of Dehn's lemma remained in doubt until 1956, when it was proven by Christos Papakyriakopoulos using his "tower construction". He also generalized the theorem to the loop theorem and sphere theorem.